Qualitative constraints and statistical methods in insurance October 2008

Qualitative constraints in insurance By Jens Perch Nielsen Professor of Actuarial Statistics Cass Business School, City University, London 2

Research method in non-life insurance RSA Cass Business School Festina Lente 3

Four succesfull research areas Estimation of large losses Operational Risk Age-period-cohort model Reserving 4

Estimation of large losses – transformation method First transformation We transform data with a cdf from the positive real axis to [0,1] Nonparametric correction The transformed data is estimated by nonparametric kernel smoothing Backtransform The distribution is backtransformed to the original positive real axis Simulations and first paper First paper was Buch-Larsen, Nielsen, Bolance, Guillen(2005). A number of papers have since appeared on variations. 5

Estimation of large losses – a qualitative constraint First the qualitative constraint On the transformed axis - [0,1] – extreme value theory tells us exactly how the density behaves in 1. Therefore, extreme value theory provides a valuable constraint useful for the transformation method Nonparametric correction The nonparametric correction has to be acknowledge the constraint from extreme value theory. Spline methods seem to better at this type of constraint than the simple kernel method One preliminary study in preparation Work in progress with Vladimir Kaishev from Cass Business School on this. 6

Operational risk The transformation method Is relevant for operational risk, where heavy tails are common Nonparametric correction Has robustifying effect Underreporting Almost all operational risk data are underreported Including external data Estimate the transformation from external data Truncation Almost all operational risk data are truncated 7

The age-period-cohort model A classical, but unsolved identifiability problem in demography But now a solution is coming out in the next Biometrika The age-period-cohort model reappears in reserving Here its called underwriting year, development year and calendar effect Claims inflation is probably one of the most urgent problems to solve And still nobody has a satisfying solution Both number of claims and payments have to be incorporated 8

Reserving - the sampling sceme First an insurance claim happens But it is not reported until later Then the insurance claim is reported But it is not being paid until later Then the insurance claim is being paid But this does not necessarily happen in one go 9

Reserving: granular data and an interesting constraint problem Classical methods only deal with aggregated data But granular data is more often available The triangle multiplicate density An interesting constrained problem Combining counts data and paid data A deconvolution problem 10